What would you do: Counter offer for a star employee?

Periodically, we ask three HR managers how they’d handle a difficult situation at work. Today’s problem: A nearly irreplaceable star performer is leaving for more money. Should you counter? 
The scene
VP Tom Harper leaned back as he spoke: “I’ll take full responsibility for the situation with Maria Diaz, but for now we should focus on making a decision about how to handle her wanting to leave.”
“I agree,” HR manager Jan Cropper nodded. She had warned Tom several times about training a replacement for Maria, but Tom had said he was always “too busy.”
“It still bothers me,” Tom said. “We hired her right out of school, invested loads of time and training in her, and now she’s leaving us for a better offer. She probably knows more about her end of the business than anyone here.”
“You have to admit, a 30% raise is a great offer from that other employer,” Jan said. “She’s done a great job here, so I guess it’s not surprising that others want her.”
“The problem now is that not only would we lose an excellent employee, but that we also won’t have anyone to replace her,” Tom sighed. “What do you think? Should we hit her with a counter offer?”
Answer within 24 hours
“If we do and word gets out here, that could cause lots of problems,” Jan noted. “If we don’t, we lose a good employee and suffer the costs of recruiting and training a replacement for her.”
“That about sums it up,” Tom concluded. “I told Maria we’d get back to her within 24 hours as to whether we would match the other offer. We have that long to decide on it.”
If you were Jan, what would you recommend to Tom?
Patty Mauro, HR manager, Winston-Salem, NC
What Patty would do: Considering the costs of replacing this person and the fact that she’s an outstanding employee, I’d work up a counter offer.
Reason: If she’s able to go out and get an offer that’s so much better and we think so highly of her, why not? I wouldn’t be too worried about other employees’ getting wind of it. My feeling is that counter offers should be done on a case-by-case basis, and I wouldn’t be too concerned about the effect on what other employees think.
Rose Suter, controller/HR director, Lancaster, PA
What Rose would do: This situation calls for a conversation with the person to learn what else she considers important besides money. Depending on what’s said in that conversation and whether we could put together working conditions that really appealed to the person, I might make a counter offer.
Reason: Rarely is the problem only about money – usually there are other issues involved. Making a counter offer may be part of the solution, but you have to make sure you can respond to the person’s other reasons for leaving, too, or else just countering will fail in the end.
Gary Lane, CEO, Cutler, ME
What Gary would do: I would consider making a counter offer, but along with it I’d work something out to move the person up to a position with greater responsibility.
Reason: It sounds as if this person hasn’t advanced at the rate she should have – if someone else thinks she’s worth 30% more, they probably also think she’s capable of taking on more. Maybe we haven’t moved her along fast enough, and she’s ready for a bigger job to go along with the bigger paycheck.